Fractional discrete Fourier transform of type IV based on the eigenanalysis of a nearly tridiagonal matrix

a Nearly Tridiagonal Matrix Magdy Tawfik Hanna1 ABSTRACT A fully-fledged definition for the fractional discrete Fourier transform of type IV (FDFT-IV) is presented and shown to outperform the simple definition of the FDFT-IV which is proved to be just a linear combination of the signal, its DFT-IV and their flipped versions. This definition heavily depends on the availability of orthonormal eigenvectors of the DFT-IV matrix G. An eigenanalysis is performed of a nearly tridiagonal matrix S which commutes with matrix G. An involutary unitary matrix P is defined and used for performing a similarity transformation that reduces S to a block diagonal form where the two diagonal blocks are exactly tridiagonal matrices. Moreover the elements of those two diagonal blocks are derived in order to circumvent the need for performing the two matrix multiplications involved in the similarity transformation. Orthonormal even and odd symmetric eigenvectors for S are generated – in terms of the eigenvectors of the two diagonal blocks – and proved to always be eigenvectors of G irrespective of the multiplicities of the eigenvalues of S. The relevance of the method contributed here is manifested in the case of a repeated eigenvalue of S with multiplicity 2 where a direct application of a general eigenanalysis procedure in any software package will not produce a pair of even and odd symmetric eigenvectors corresponding to this repeated eigenvalue. It should be mentioned that the almost tridiagonal matrix S which commutes with the DFT-IV matrix G being dealt with here is distinct from matrix S which commutes with the DFT matrix F dealt with in a previous paper [7].